The volume of a rectangular prism is
. What are the lengths of the prism's sides?

Guidance

The last type of factorable polynomial are those that are in quadratic form.
Quadratic form
is when a polynomial looks like a trinomial or binomial and can be factored like a quadratic. One example is when a polynomial is in the form
. Another possibility is something similar to the difference of squares,
. This can be factored to
or
. Always keep in mind that the greatest common factors should be factored out first.

Example A

Factor
.

Solution:
This particular polynomial is factorable. Let’s use the method we learned in the
Factoring when
concept. First,
. The factors of -30 that add up to -1 are -6 and 5. Expand the middle term and then use factoring by grouping.

Both of the factors are not factorable, so we are done.

Example B

Factor
.

Solution:
Treat this polynomial equation like a difference of squares.

Now, we can factor
using the difference of squares a second time.

cannot be factored because it is a sum of squares. This will have imaginary solutions.

Example C

Find all the real-number solutions of
.

Solution:
First, pull out the GCF among the three terms.

Factor what is inside the parenthesis like a quadratic equation.
and the factors of -18 that add up to -17 are -18 and 1. Expand the middle term and then use factoring by grouping.

Factor
further and solve for
where possible.
is not factorable.

Intro Problem Revisit
To find the lengths of the prism's sides, we need to factor
.

First, pull out the GCF among the three terms.

Factor what is inside the parenthesis like a quadratic equation.
and the factors of -6 that add up to -5 are -6 and 1.

Therefore, the lengths of the rectangular prism's sides are
,
, and
.

Guided Practice

Factor the following polynomials.

1.

2.

3. Find all the real-number solutions of
.

Answers

1.
and the factors of 24 that add up to 14 are 12 and 2.

2. Factor this polynomial like a difference of squares.

6 and 5 are not square numbers, so this cannot be factored further.

3. Pull out a
from each term.

Set each factor equal to zero.

Notice the second factor will give imaginary solutions.

Vocabulary

Quadratic form

When a polynomial looks a trinomial or binomial and can be factored like a quadratic equation.

Practice

Factor the following quadratics completely.

Find all the real-number solutions to the polynomials below.

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Description

This lesson applies factoring a quadratic equation to other types of polynomials that are in "quadratic form."

Learning Objectives

Here you'll how to factor and solve polynomials that are in “quadratic form.”